Stress-energy tensor of a perfect fluid

[notknowing]

The stress energy tensor of a perfect fluid is composed of two terms of which only one term contains the metric tensor gab. (product of metric tensor and pressure). For curved spacetime, one replaces the flat spacetime metric tensor by the metric tensor of curved space. What I find bizar however is that the metric tensor enters into the expression of the stress-energy tensor in such an asymmetric way (one part affected, the other not). The same applies for the stress energy tensor of the electromagnetic field. Is there some deep reason why one part is affected while the other part is not ? Should one not expect that the geometry of spacetime affects all components in a similar way ?

 

[pmb_phy]

The stress energy tensor of a perfect fluid is composed of two terms of which only one term contains the metric tensor gab. (product of metric tensor and pressure).

Huh? Where did you get that from? The stress-energy-momentum for a perfect fluid is

T^{\alpha\beta} = \rho U^{\alpha} U^{\beta}

(I used "tex" in square brackets but got a latex error. What did I do wrong??)

where U is the 4-momentum of a fluid element

For curved spacetime, one replaces the flat spacetime metric tensor by the metric tensor of curved space.

If you're talking about the components of the tensor at a given point then you can choose whatever system of coordinates you like, even locally a locally flat coordinate system in which the metric tensor is diag(1,-1,-1,-1).

Best wishes

Pete

 

[MeJennifer]

Huh? Where did you get that from? The stress-energy-momentum for a perfect fluid is

T^{\alpha\beta} = \rho U^{\alpha} U^{\beta}

In the wikipedia article http://en.wikipedia.org/wiki/Fluid_solution" it is written as:

$$T^{ab} = (\mu + p) \, u^a \, u^b + p \, g^{ab}$$

I think he is asking why the velocity vectors are not subject to the metric as the pressure is.

By the way, happy birthday Pete!

 

[notknowing]

In the wikipedia article http://en.wikipedia.org/wiki/Fluid_solution" it is written as:

$$T^{ab} = (\mu + p) \, u^a \, u^b + p \, g^{ab}$$

I think he is asking why the velocity vectors are not subject to the metric as the pressure is.

By the way, happy birthday Pete!

Indeed, my question is why the vectors are not subject to the metric as the pressure is.

 

[pmb_phy]

By the way, happy birthday Pete!

Thank you Jen! Much appreciated. By the way, how did you know it was my birthday today?

The Birthday Boy Pete

ps - My previous post was incorrect. I posted the tensor for a dust. Thanks Jen for getting it right and correcting me.

 

[Stingray]

You're essentially splitting the stress-energy tensor into two parts that are orthogonal to each other. Since the density portion has the tensor structure $u^{a} u^{b}$, the pressure term should be proportional to a rank-2 symmetric tensor that leaves zero when contracted with the 4-velocity. But contracting it with any vector orthogonal to u will just give you that vector back (with a raised index). It's easy to verify that the only object satisfying these requirements is the standard projection operator
$$g^{ab} + u^{a} u^{b}$$

Edit: TeX doesn't seem to be working right now. The first bit is supposed to read u^{a} u^{b}, and the second should say g^{ab}+u^{a}u^{b}.

 

[notknowing]

You're essentially splitting the stress-energy tensor into two parts that are orthogonal to each other. Since the density portion has the tensor structure $u^{a} u^{b}$, the pressure term should be proportional to a rank-2 symmetric tensor that leaves zero when contracted with the 4-velocity. But contracting it with any vector orthogonal to u will just give you that vector back (with a raised index). It's easy to verify that the only object satisfying these requirements is the standard projection operator
$$g^{ab} + u^{a} u^{b}$$

Edit: TeX doesn't seem to be working right now. The first bit is supposed to read u^{a} u^{b}, and the second should say g^{ab}+u^{a}u^{b}.

I understand that there are clear and strict mathematical rules to construct the tensors with the right properties but is there some way to understand this from a more physical point of view ? How is it possible that part of the tensor (the uaub term) is completely independent of the metric ? Maybe it is not possible to have a better understanding than the mathematical point of view. I was just wondering ..

 

[Stingray]

I understand that there are clear and strict mathematical rules to construct the tensors with the right properties but is there some way to understand this from a more physical point of view ? How is it possible that part of the tensor (the uaub term) is completely independent of the metric ? Maybe it is not possible to have a better understanding than the mathematical point of view. I was just wondering ..

How could it involve the metric? In a sense, u^{a} involves it implicitly since it is supposed to be normalized. But beyond that, any terms would just be absorbed into \rho.

 

[notknowing]

How could it involve the metric? In a sense, u^{a} involves it implicitly since it is supposed to be normalized. But beyond that, any terms would just be absorbed into \rho.

Sorry for the late reply but more down to Earth problems have occupied me
My knowledge of mathematics and relativity is not so advanced as yours, so I have some difficulty in following your arguments. I do not understand how the metric is implicitly involved in u^{a} (by normalisation). Could you explain this in some more detail ?

 

[Stingray]

Sorry for the late reply but more down to Earth problems have occupied me
My knowledge of mathematics and relativity is not so advanced as yours, so I have some difficulty in following your arguments. I do not understand how the metric is implicitly involved in u^{a} (by normalisation). Could you explain this in some more detail ?

When writing the $\rho u^{a} u^{b}$, it is assumed that $u^{a} u_{a} = g_{ab} u^{a} u^{b} = -1$. You cannot make this statement without using the metric. If it isn't imposed, $\rho$ loses its meaning.

 

[notknowing]

When writing the $\rho u^{a} u^{b}$, it is assumed that $u^{a} u_{a} = g_{ab} u^{a} u^{b} = -1$. You cannot make this statement without using the metric. If it isn't imposed, $\rho$ loses its meaning.

Thanks for this quick reply. I got your point. Suppose one would multiply the metric by some constant (scalar), how would this change, in general, the stress energy tensor of matter, fields, etc. ?

 

[George Jones]

It's also illustrative to consider Stingray's last point using index-free notation, i.e., consider the stress-energy tensor to be a bilinear map that takes pairs of 4-vectors into scalars. The stress-energy tensor for a fluid acting on arbitrary 4-vectors $v$ and $w$ gives

$$T \left( v , w \right) = \left( \rho + p \right) g \left( v , u \right) g \left( w , u \right) + pg \left( v , w \right).$$

 

[notknowing]

It's also illustrative to consider Stingray's last point using index-free notation, i.e., consider the stress-energy tensor to be a bilinear map that takes pairs of 4-vectors into scalars. The stress-energy tensor for a fluid acting on arbitrary 4-vectors $v$ and $w$ gives

$$T \left( v , w \right) = \left( \rho + p \right) g \left( v , u \right) g \left( w , u \right) + pg \left( v , w \right).$$

Thanks George. Coming back to my last question ; If one would multiply the metric tensor by some scalar constant A, how would the stress energy tensor of a perfect fluid, or the stress energy tensor in general, be modified ?

 

[Stingray]

Thanks for this quick reply. I got your point. Suppose one would multiply the metric by some constant (scalar), how would this change, in general, the stress energy tensor of matter, fields, etc. ?

Sorry I forgot to answer before. Anyway, what you're asking is a special case of something known as a conformal transformation (or conformal isometry). Multiplying the metric by a positive constant $A^{2}$ is equivalent to a coordinate transformation $x \rightarrow x/A$. The stress-energy tensor would then transform as $T^{ab} \rightarrow T^{ab}/A^{2}$.

 

[notknowing]

Sorry I forgot to answer before. Anyway, what you're asking is a special case of something known as a conformal transformation (or conformal isometry). Multiplying the metric by a positive constant $A^{2}$ is equivalent to a coordinate transformation $x \rightarrow x/A$. The stress-energy tensor would then transform as $T^{ab} \rightarrow T^{ab}/A^{2}$.

Thanks Stingray for this very interesting remark. How did you work out this so quickly ?
What I'm actually trying to find out is how the Einstein Field equations transform when the metric tensor is multiplied by a constant or in other words, how would the Einstein tensor change under such a transformation ?

 

[Stingray]

Thanks Stingray for this very interesting remark. How did you work out this so quickly ?
What I'm actually trying to find out is how the Einstein Field equations transform when the metric tensor is multiplied by a constant or in other words, how would the Einstein tensor change under such a transformation ?

By the (old fashioned) definition of tensor, a coordinate transformation $\bar{x}^{a}=\bar{x}^{a}(x)$ transforms the components of any tensor with two upper indices as

$$\bar{L}^{ab} = \frac{ \partial \bar{x}^{a} }{ \partial x^{c} } \frac{ \partial \bar{x}^{b} }{ \partial x^{d} } L^{cd}$$

Similarly,

$$\bar{L}_{ab} = \frac{ \partial x^{c} }{ \partial \bar{x}^{a} } \frac{ \partial x^{d} }{ \partial \bar{x}^{b} } L_{cd}$$

This applies to the Einstein and metric tensors just like it does to any other.

If the factor that you're multiplying the metric by isn't a constant, then the interpretation as a coordinate transformation isn't always possible. But it can still be a useful thing to do, and is called a conformal transformation. It is described in detail in one of the appendices in Wald's textbook.

 

[notknowing]

By the (old fashioned) definition of tensor, a coordinate transformation $\bar{x}^{a}=\bar{x}^{a}(x)$ transforms the components of any tensor with two upper indices as

$$\bar{L}^{ab} = \frac{ \partial \bar{x}^{a} }{ \partial x^{c} } \frac{ \partial \bar{x}^{b} }{ \partial x^{d} } L^{cd}$$

Similarly,

$$\bar{L}_{ab} = \frac{ \partial x^{c} }{ \partial \bar{x}^{a} } \frac{ \partial x^{d} }{ \partial \bar{x}^{b} } L_{cd}$$

This applies to the Einstein and metric tensors just like it does to any other.

If the factor that you're multiplying the metric by isn't a constant, then the interpretation as a coordinate transformation isn't always possible. But it can still be a useful thing to do, and is called a conformal transformation. It is described in detail in one of the appendices in Wald's textbook.

So, if I understand it correctly, the Einstein Field Equations ARE invariant under a multiplication of the metric tensor by a constant. But would this invariance not be spoiled by adding the cosmological constant term ? And would this not be an argument against the introduction of this term ?

 

[Stingray]

So, if I understand it correctly, the Einstein Field Equations ARE invariant under a multiplication of the metric tensor by a constant. But would this invariance not be spoiled by adding the cosmological constant term ? And would this not be an argument against the introduction of this term ?

Multiplying the metric by a constant is equivalent to a coordinate transformation. Einstein's equations (with or without the cosmological constant) are invariant under all coordinate transformations.

 

[notknowing]

Multiplying the metric by a constant is equivalent to a coordinate transformation. Einstein's equations (with or without the cosmological constant) are invariant under all coordinate transformations.

Something must be wrong here. If you multiply the metric by a constant, then the cosmological constant term will also be multiplied by that same constant, while the stress energy tensor will be divided by that constant (your previous messages). So, clearly they do not scale in the same way and so Einsteins equations can not be invariant under such a change.

 

[pervect]

I don't understand what the problem is. Stingray is right, multiplying by a constant is just a coordinate transformation. Perhaps you need to set up a specific example. Suppose we have a metric g, and a metric gg, representing the same space-time, and all of the coefficients of gg are 4 times as large as the coefficeints of g.

Then ds^2 = g_ij dx^i dx^j = gg_ij dxx^i dxx^j

where gg_ij = 4 g_ij
xx^i = (1/2) x^i

So when you label all coordinates x^i with 1/2 their old labels, you increase the metric coefficeints g_ij by a factor of 4

 

[notknowing]

I don't understand what the problem is. Stingray is right, multiplying by a constant is just a coordinate transformation. Perhaps you need to set up a specific example. Suppose we have a metric g, and a metric gg, representing the same space-time, and all of the coefficients of gg are 4 times as large as the coefficeints of g.

Then ds^2 = g_ij dx^i dx^j = gg_ij dxx^i dxx^j

where gg_ij = 4 g_ij
xx^i = (1/2) x^i

So when you label all coordinates x^i with 1/2 their old labels, you increase the metric coefficeints g_ij by a factor of 4

When I asked how the energy-stress tensor (T) changed under a multiplication by a constant (say B), Stingray answered (see 10-29-2006 , 10:53) that it transformed to T/B. If you transform the cosmological constant term (by multiplying g by B), then this term is obviously multiplied by B. Dividing by B is not the same as multiplying by B, so the two terms (cosmological constant term and T) do not trasnform in the same way. This is the problem.

 

[Stingray]

When I asked how the energy-stress tensor (T) changed under a multiplication by a constant (say B), Stingray answered (see 10-29-2006 , 10:53) that it transformed to T/B. If you transform the cosmological constant term (by multiplying g by B), then this term is obviously multiplied by B. Dividing by B is not the same as multiplying by B, so the two terms (cosmological constant term and T) do not trasnform in the same way. This is the problem.

It makes a difference whether the indices are up or down. It is conventional (though not necessary) to write the metric with lowered indices, so I was considering the transformation $g_{ab} \rightarrow B g_{ab}$. But that means that $g^{ab} \rightarrow g^{ab}/B$. Similarly, stress-energy tensors are usually written with raised indices, so that's what I mentioned: $T^{ab} \rightarrow T^{ab}/B$. But if you look at

$$G_{ab} + \Lambda g_{ab} = 8 \pi T_{ab}$$

each term will be multiplied by B under your transformation.

 

[pmb_phy]

When I asked how the energy-stress tensor (T) changed under a multiplication by a constant (say B), Stingray answered (see 10-29-2006 , 10:53) that it transformed to T/B. If you transform the cosmological constant term (by multiplying g by B), then this term is obviously multiplied by B. Dividing by B is not the same as multiplying by B, so the two terms (cosmological constant term and T) do not trasnform in the same way. This is the problem.

First off, multiplying by a constant (I assume you're referring to a scalar) is not a coordinate transformation. E.g. recall 3d vector analysis. Let's use the vector R = (a, b, c). Now multiply by the constant q to give qR = (qa, qb, qc). This is simply another vector whose length is q times the length of the original vector. Same thing applies to tensors of any rank.

And the result is not T/B, its BT.

Best wishes

Pete

 

[Stingray]

First off, multiplying by a constant (I assume you're referring to a scalar) is not a coordinate transformation. E.g. recall 3d vector analysis. Let's use the vector R = (a, b, c). Now multiply by the constant q to give qR = (qa, qb, qc). This is simply another vector whose length is q times the length of the original vector. Same thing applies to tensors of any rank.

And the result is not T/B, its BT.

Read what I wrote about it. It is correct. He was talking about multiplying the metric by a constant scalar. That is most easily visualized as a coordinate transformation. You can get "inverse scaling" depending on whether the object is covariant or contravariant.

 

[Chris Hillman]

Perfect fluids in gtr

Hi, notknowing,

The stress energy tensor of a perfect fluid is composed of two terms of which only one term contains the metric tensor gab. (product of metric tensor and pressure). For curved spacetime, one replaces the flat spacetime metric tensor by the metric tensor of curved space. What I find bizar however is that the metric tensor enters into the expression of the stress-energy tensor in such an asymmetric way (one part affected, the other not). The same applies for the stress energy tensor of the electromagnetic field. Is there some deep reason why one part is affected while the other part is not ? Should one not expect that the geometry of spacetime affects all components in a similar way ?

Indeed, my question is why the vectors are not subject to the metric as the pressure is.

I am coming into this thread rather late, but for what it is worth, I think I know what you mean, but you were simply being confused by the notation. I think the best way to understand this expression is to study the discussion of perfect fluids in the textbooks by Schutz or D'Inverno, until you understand how it is derived.

Stingray made a remark about scaling, an notknowing inquired:

Thanks Stingray for this very interesting remark. How did you work out this so quickly ?

I think the reading I recommended will clarify this (think of how a velocity is scaled, then note that the contribution to the stress-energy tensor from a fluid is quadratic in velocities).

The stress-energy-momentum for a perfect fluid is
$T^{\alpha\beta} = \rho \, U^{\alpha} \, U^{\beta}$
where U is the 4-momentum of a fluid element

Actually, this expression represents the contribution from a pressureless perfect fluid or "dust". See for example (21.1) in D'Inverno's textbook.

I am pretty sure that notknowing was referring to the case of nonzero pressure,
$T^{ab} = \left( \rho + p \right) \, U^a U^b - p \, g^{ab}$
where p is the pressure and $$\rho$$ is the density, both measured by an idealized observer comoving with a fluid particle. (George Jones has already quoted this expression written in slightly more modern notation.)

First off, multiplying by a constant (I assume you're referring to a scalar) is not a coordinate transformation. E.g. recall 3d vector analysis. Let's use the vector R = (a, b, c). Now multiply by the constant q to give qR = (qa, qb, qc). This is simply another vector whose length is q times the length of the original vector. Same thing applies to tensors of any rank.

Rescaling the coordinates is called a "dilation". This is an example of a conformal transformation, and of course it is also a diffeomorphism, and thus any tensor equation will therefore be invariant under a dilation. However, individual COMPONENTS will rescale, which seems to be the basis for the confusion in this thread (I think several participants might have been talking past one another). Indeed, thinking in terms of dimensional analysis for vectorial quantities in Euclidean space should give the right idea for how tensor components scale in curved spacetimes--- it seems to me that Peter and Stingray actually agree about this point.

Chris Hillman